To solve the system of equations, we’ll analyze step-by-step:

The system is:

1. $2x - 3y - 9z = -24$
2. $x + 3z = 0$
3. $-3x + y - 4z = 8$

Step 1: Express $x$ from the second equation

From $x + 3z = 0$, we solve for $x$: $x = -3z$

Step 2: Substitute $x = -3z$ into the first and third equations

Substituting into $2x - 3y - 9z = -24$:

$2(-3z) - 3y - 9z = -24$ $-6z - 3y - 9z = -24$ $-15z - 3y = -24$ Divide through by $-3$: $5z + y = 8 \quad \text{(Equation 4)}$

Substituting into $-3x + y - 4z = 8$:

$-3(-3z) + y - 4z = 8$ $9z + y - 4z = 8$ $5z + y = 8 \quad \text{(Equation 5, same as Equation 4)}$

Step 3: Analyze the system

From both substitutions, we see that $5z + y = 8$ holds true. This implies there is no contradiction, and the system is dependent.

Step 4: Solve for $y$ in terms of $z$

From $5z + y = 8$, solve for $y$: $y = 8 - 5z$

Step 5: Express the solution

Since $x = -3z$, $y = 8 - 5z$, and $z = z$ (free variable), the solution can be written as: $x = -3z, \quad y = 8 - 5z, \quad z = z$

Final Answer

(a) The system is dependent.
(b) The solution is: $x = -3z, \quad y = 8 - 5z, \quad z = z$

Let me know if you'd like more details or clarifications! Here are 5 related questions to expand your understanding:

1. How can you verify if a system is consistent, inconsistent, or dependent using matrices?
2. What does it mean geometrically for a system to be dependent?
3. How would the solution change if one equation were inconsistent?
4. Can a dependent system still have unique solutions in some cases? Why or why not?
5. How would substitution differ if the second equation had no $z$-term?

Tip: For dependent systems, always look for free variables to express the solutions parametrically.